Problem: Let $h(x)=6\log(x)$. Find $h'(x)$. Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac{6}{x\ln(10)}$ (Choice B) B $\dfrac{6}{x\ln(x)}$ (Choice C) C $\dfrac{6}{x}$ (Choice D) D $\dfrac{6}{\log(x)}$
Solution: The expression for $h(x)$ includes a logarithmic term. Remember that the derivative of the general logarithmic term $\log_a(x)$ (where $a$ is any positive constant and $a\neq 1$ ) is $\dfrac{1}{\ln(a)\cdot x}$. Put another way, $\dfrac{d}{dx}[\log_a(x)]=\dfrac{1}{\ln(a)\cdot x}$. [Is there an easy way to memorize that?] We can use this to find the derivative of the function as shown below. $\begin{aligned} h(x)&=\dfrac{d}{dx}[6\log(x)] \\\\ &=6\dfrac{d}{dx}[\log(x)] \\\\ &=6\dfrac{d}{dx}[\log_{10}(x)]&&{\gray{\text{Since }\log(x)=\log_{10}(x)}} \\\\ &=6\cdot\dfrac{1}{\ln(10)x} \\\\ &=\dfrac{6}{x\ln(10)} \end{aligned}$ In conclusion, $h'(x)=\dfrac{6}{x\ln(10)}$.